Question: The equation of hyperbola $H$ is $\dfrac {(x+8)^{2}}{49}-\dfrac {(y+9)^{2}}{16} = 1$. What are the asymptotes?
Solution: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+9)^{2}}{16} = - 1 + \dfrac {(x+8)^{2}}{49}$ Multiply both sides of the equation by $16$ $(y+9)^{2} = { - 16 + \dfrac{ (x+8)^{2} \cdot 16 }{49}}$ Take the square root of both sides. $\sqrt{(y+9)^{2}} = \pm \sqrt { - 16 + \dfrac{ (x+8)^{2} \cdot 16 }{49}}$ $ y + 9 = \pm \sqrt { - 16 + \dfrac{ (x+8)^{2} \cdot 16 }{49}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 9 \approx \pm \sqrt {\dfrac{ (x+8)^{2} \cdot 16 }{49}}$ $y + 9 \approx \pm \left(\dfrac{4 \cdot (x + 8)}{7}\right)$ Subtract $9$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{4}{7}(x + 8) -9$